Niagara Falls, December 2 - 5, 2016
The theoretical framework that we have developed to study intersections of Cantor sets, including the idea of a path set and of $p$-adic path set fractals, has found application to the study of multi-layer cellular networks. I will also discuss current work using this framework to study the self-similarity of intersections and unions of translations of Cantor sets.
The talk is based on my joint works with Antti K\"aenm\"aki, Henna Koivusalo, Micha\l\ Rams and K\'aroly Simon.
In joint work with M. Ionescu and K. Okoudjou we consider an analogue of this result on the Sierpinski Gasket in which $[f]$ is replaced by a pseudo-differential operator, the projection is onto the eigenspaces of $L^2$ with eigenvalues less than $\Lambda$ and the limit is taken as $\Lambda\to\infty$. For a suitable class of pseudo-differential operators the result gives asymptotics of the operator from its symbol.
Keywords: Fractal, Transport properties, Porous media.
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